Riemannian Optimization for Variance Estimation in Linear Mixed Models

TL;DR

This paper introduces a Riemannian geometric approach to variance estimation in linear mixed models, improving the quality of estimates by leveraging the intrinsic geometry of the parameter space.

Contribution

It formulates variance estimation as an optimization problem on a Riemannian manifold, providing higher-order geometric information and demonstrating improved estimation accuracy.

Findings

Higher quality variance estimates compared to existing methods

Effective use of Riemannian gradient and Hessian in optimization

Numerical validation shows improved performance

Abstract

Variance parameter estimation in linear mixed models is a challenge for many classical nonlinear optimization algorithms due to the positive-definiteness constraint of the random effects covariance matrix. We take a completely novel view on parameter estimation in linear mixed models by exploiting the intrinsic geometry of the parameter space. We formulate the problem of residual maximum likelihood estimation as an optimization problem on a Riemannian manifold. Based on the introduced formulation, we give geometric higher-order information on the problem via the Riemannian gradient and the Riemannian Hessian. Based on that, we test our approach with Riemannian optimization algorithms numerically. Our approach yields a higher quality of the variance parameter estimates compared to existing approaches.